Non-myopic Informative Path Planning in Spatio-Temporal Models Alexandra Meliou Andreas Krause Carlos Guestrin Joe Hellerstein

Collection Tours

Approximate Queries Approximate representation of the world: Discrete locations Lossy communication Noisy measurements Applications do not expect accurate values (tolerance to noise) Monitored phenomena usually demonstrate strong correlations Correlation makes approximation cheap Example: Return the temperature at all locations ±1C, with 95% confidence

Optimizing Information : sensing nodes on path Approximate answers Search for most informative paths

Continuous Queries Repeated at periodic intervals Finite horizon Example: Return the temperature at all locations ±1C, with 95% confidence, every 10 minutes for the next 5 hours.

Myopic vs Nonmyopic tradeoff Myopic approach: repeat optimization for every timestep Timestep 1Timestep 2

Myopic vs Nonmyopic tradeoff Nonmyopic approach: optimize for all timesteps Timestep 1Timestep 2 No work! Extra node

Quantify Informativeness Entropy [Shewry & Wynn ‘87] Mutual Information [Caselton & Zidek ‘84] Reduction of predictive variance [Chaloner & Verdinelli ‘95]

Measuring Information Observing 1 gives information on 3 and 4 Observing 2 gives information on 3 and 5 After observing 2, observing 3 becomes less useful Diminishing Returns

Submodular Functions B A X X + + More reward Less reward Entropy, mutual information and reduction of predictive variance are all submodular.

Non-myopic Spatio-Temporal Path Planning (NSTP) Given: A collection of submodular functions f t f t only depends on data collected at times 1..t A set of accuracy constraints k t Find: A collection of paths P t with Minimize cost Subject to reward constraints

Planning for multiple timesteps Harder than planning for one First idea : Solve an equivalent single step problem instead! obviously

Nonmyopic Planning Graph t=1 t=2 t=3 A solution path on the NPG = collection of paths for multiple timesteps

Solve the single step problem NP hard No good known approximation guarantees Dual: Submodular Orienteering Problem dual:primal: Maximize reward Subject to budget constraints Minimize cost Subject to reward constraints

Good News The dual algorithm [Chekuri & Pal ’05] provides an O(logn) factor approximation (where n is the size of the network)

Covering Algorithm Transform a dual blackbox solution to a primal solution dual: primal: Reward required to “cover” (with α approximation factor) Call with B OPT Return solution with reward ≥K/α

Covering Algorithm Transform a dual blackbox solution to a primal solution Reward required to “cover” Call SOP for increasing budgets Guaranteed to cover K/α reward when called for B OPT Update chosen set and repeat for uncovered reward Terminate when ε portion left Guaranteed to use at most budget Call for budget 1 : insufficient reward Call for budget 2 Call for budget B OPT : reward sufficient! uncovered reward

Bad News On the unrolled graph the Chekuri-Pal guarantee becomes O(log(nT)) The running time on the unrolled graph is O((BnT) log(nT) )

Addressing Computation Complexity DP Algorithm Algorithm details in proceedings Bug in proof of guarantees. Not fixed (yet) New algorithm: Nonmyopic Greedy Details on my webpage… Guaranteed to provide O(logn) approximation Better than the previous O(log(nT))

Approach Replace expensive blackbox, with cheaper blackbox Covering transformation Chekuri-Pal SOP on NPG Blackbox for dual Nonmyopic greedy algorithm Blackbox for dual More efficient: Nonmyopic greedy calls the dual on the smaller network graph instead of the unrolled graph

Nonmyopic Greedy Time Budget dual(b,G t ) R = 2 C = 1 R = 1 C = 1 R = 1 C = 1 R = 3 C = 2 R = 5 C = 4 R = 4 C = 2 R = 3 C = 2 R = 6 C = 4 R = 3 C = 2 R = 5 C = 3 R = 4 C = 3 R = 5 C = 4 budget P 1 Cost = 2 Time = 2 Best greedy choice condition on A 1 A2A2 R = 2 C = 1 R = 1 C = 1 R = 1 C = 1 R = 2 C = 2 X R = 1 C = 2 R = 1 C = 2 X XXX X P 2 Cost = 1 Time = 1 R = 0 C = 1 R = 0 C = 1 R = 1 C = 1 XXX P 3 Cost = 1 Time = 3 Best ratio R/C 1. Condition on picked data 2. Recompute matrix A1A1 Return best of A 1, A 2 dual(budget=4,time=1) dual(budget=1,time=3) For border cases were A 1 is bad, A 2 is guaranteed to be good

Nonmyopic Greedy Guarantees Nonmyopic greedyChekuri-Pal on NPG O(B 2 T(nB) logn )O((nBT) log(nT) ) running time approximation

Myopic and Nonmyopic evaluation Varying Constraints Setup:  46 nodes on the Intel Berkeley Lab deployment  7 days of data (5 for learning, 2 for testing)

Cost and Runtime Varying Horizon

Effect of greedy parameters Varying budget levels

Conclusions Transform any blackbox solution to nonmyopic Obtain primal from dual Nonmyopic greedy provides significant runtime improvements and better theoretical guarantees